Determine the solvability conditions for the nonhomogeneous boundralue problem: \(u^{\prime}(\pi / 4)=\beta\).

Short Answer

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The nonhomogeneous boundary value problem \(u^{\prime}( \pi / 4) = \beta\) is solvable for any value of \(\beta\).

Step by step solution

01

Understanding the equation

The given equation \(u^{\prime}( \pi / 4) = \beta\) represents the derivative of the function u at the point \(\pi / 4\), which is equal to a constant \(\beta\). This is a first-order differential equation.
02

Determining the General solution

The general solution to this kind of first-order differential equation is of the form \(u(x) = \beta x + C\), where C is a constant. This is because the derivative of this function gives a constant, \(\beta\) in this case.
03

Applying the boundary condition

Apply the boundary condition to the general solution to solve for C. Here, the boundary value is given at \(x = \pi / 4\). Substituting \(x = \pi / 4\) into the general solution will yield \(u( \pi / 4) = \beta (\pi / 4) + C\). However, we are not given a certain value for \(u\), hence, the unique solution cannot be determined. Thus, instead, the problem is solvable for any value of \(\beta\).

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